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This is a rather simple question intuitively, but I can't seem to find a rigorous explanation for this: suppose we are diving a rectangle into some number of equal sized squares, then they must all be aligned together(such that we can produce theses squares by performing horizontal and vertical cuts on the rectangle).

It makes sense to me, since if we have misalignment, then eventually we are going to reach a position where we can't fit the square. However, this assumes that the squares are originally aligned in the first place... Any rigorous explanation/counter example to this?

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  • $\begingroup$ You should always be able to tesselate a rectangle with equally-sized squares as long as the ratio of length to width is a rational number. A counter example would be a rectangle with length $\sqrt{2}$ and width $1$. $\endgroup$ – D.B. May 25 '18 at 19:11
  • $\begingroup$ To be clear -- you're asking if there's a way to tile a rectangle with squares other than a regular pattern? $\endgroup$ – dbx May 25 '18 at 19:19
  • $\begingroup$ @dbx yes, I was wondering if there's a way to tile a rectangle with equal sized squares other than the regular grid pattern, but it seems that there will be no such ways since the answers suggest to orient the squares in the corners and then the right angles will force the squares to be in a grid. $\endgroup$ – Jesse Meng May 25 '18 at 19:47
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I think it should be obvious given that you need to put a square in the corner in the rectangle (if you move it a distance away from one of the sides of the corner, that is less than the length of the square, you'll leave an unfillable gap). And, with that first square in place, you just created a new corner to be filled up, and again you need to put a square exactly in that spot ... etc. This way, it should be clear that as you go along one side of the rectangle, you need to align all squares, and once that first row is done, the same story applies to the next row, etc. .. until the rectangle is all tiled.

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Sketch: A corner which is a right-angle forces an orientation for the square which fills it. Once you run out of such corners you have filled the rectangle. Therefore all the squares are forced to have the same orientation.

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  • $\begingroup$ thank you, this makes a lot of sense to me. $\endgroup$ – Jesse Meng May 25 '18 at 19:48

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